Scores of U.S. charter schools run by Turkish-Americans risk getting caught up in the political fallout between the U.S. and Turkey over a cleric living in the U.S. but accused of orchestrating Friday’s failed coup in Istanbul.

The Turkish government has threatened to make the U.S. an enemy if it does not turn over Fethullah Gulen, who has lived a reclusive life of self-imposed exile in a compound in Saylorsburg, Pa., about 90 miles north of Philadelphia, for nearly two decades.

Mr. Gulen denies having any role in the military’s attempt to unseat democratically elected President Recep Tayyip Erdogan last week. Instead, Mr. Gulen accuses Mr. Erdogan himself of orchestrating the coup to legitimize his own rule.

The cleric’s international following, known as the Gulen movement, is said to be tied to around 150 U.S. charter schools, ranging from networks in Texas, Illinois and Florida to stand-alone academies in Maryland, said Joshua Hendrick, an assistant professor of sociology and global studies at Loyola University Maryland, who studies the Gulen movement.

The political controversy won’t have any immediate effect on the schools, since Mr. Gulen does not have a direct hand in operating them, and many of the schools dispute any connection to the cleric, Mr. Hendrick said.

But in the long term, the accusations against Mr. Gulen and tension between the U.S. and Turkey could affect new schools and charters up for renewal that are run by Turkish-Americans and are said to be connected with the cleric, Mr. Hendrick said.

“Charters are decided by school boards and school boards are driven by American political culture,” said Mr. Hendrick. “That’s going to directly affect the political perspective of the people voting for a charter school proposal or renewal.”

Local school boards may be hesitant to approve schools affiliated with a man who is accused of trying to overthrow a foreign government, especially if Mr. Gulen is extradited, said Mr. Hendrick.